On the efficiency of exchange in parallel tempering Monte Carlo simulations

We introduce the concept of effective fraction, defined as the expected probability that a configuration from the lowest index replica successfully reaches the highest index replica during a replica exchange Monte Carlo simulation. We then argue that the effective fraction represents an adequate measure of the quality of the sampling technique, as far as swapping is concerned. Under the hypothesis that the correlation between successive exchanges is negligible, we propose a technique for the computation of the effective fraction, a technique that relies solely on the values of the acceptance probabilities obtained at the end of the simulation. The effective fraction is then utilized for the study of the efficiency of a popular swapping scheme in the context of parallel tempering in the canonical ensemble. For large dimensional oscillators, we show that the swapping probability that minimizes the computational effort is 38.74%. By studying the parallel tempering swapping efficiency for a 13-atom Lennard-Jones cluster, we argue that the value of 38.74% remains roughly the optimal probability for most systems with continuous distributions that are likely to be encountered in practice.